,so in class we're covering second-order D.E.s... How exactly was it determined/derived that all solutions to homogeneous 2nd-order linear differential equations with constant coefficients (ay'' + by' + cy =0, where a,b,c are real constants) are of the form y=ert. Not asking for anyone to "do [my] homework" for me - I'm just curious. In lecture, and on the professor's website it simply says, "The reason for this is that long ago some geniuses figured this stuff out and it works." immediately after introducing the form of the solutions. ,also while I'm on the topic I have a question as to WHY we should continue considering the linear combination of solutions to these equations? ..ie so suppose we're given y''-y=0 ,with a little work we can see that y1=2et and y2=5e-t are two possible solutions to the D.E. ,and it can be seen that c1y1+c2y2=y is also a solution ,why should/do we continue to only concentrate on the linear combination, when we could simply return to the multiplies of solutions y1 and y2 to solve any general or initial value problem?